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Allocation Before the Clock: Graph, Set and Game Theory of Professional League of Legends Objective Setups

Zheng Gao, Zhenyu Liu, Yifan Yang, Xiaoyu Li

arXiv:2610.33626v1cs.CVstat.AP

TL;DR

Professional teams make largely unmeasured choices about player positioning and commitment before neutral objectives. The paper models those choices as allocations on a walled map and estimates an equilibrium; it finds trailing teams more often leave the pit empty, setup patterns are reproducible, and teams weight fights, travel, and player costs more than objective value.

  • Problem

    Professional teams plan player positioning and commitment before neutral objectives, but esports analytics has measured almost none of these setups.

    Source: §1 · p. 2

  • Method

    The paper represents setups as allocations on a walled navigation graph and combines partition-lattice pattern discovery with an allocation game priced in win probability.

    Source: §1 · p. 2

  • Results

    +34.8 pp [18.7, 50.0] at the first baron: trailing teams leave the pit empty more often, while HASSE whole-model refits agree at ARI 0.819.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Teams price fights, travel, and committed players more than the objective itself, and the equilibrium reproduces about a quarter of the commitment gap, almost entirely through state variables.

    Source: §1 · pp. 2–3

    The data cover one rule set and two international events, and the model leaves much of the commitment gap unexplained by its strategic channels.

    Source: §7 · p. 16

Abstract

from arXiv · show

Before every neutral objective in professional League of Legends, each team decides which of its five players walk to the pit, which hold a lane, and when to commit. The leading analysis of such setups, TAGNet for Counter-Strike 2, clusters graph-network embeddings of player positions and reads cluster win rates as tactical quality. We treat a setup as an allocation on a walled map: formations are set partitions on the lattice $\Pi_5$, control is a geodesic Voronoi split of a navigation graph, and commitment is an allocation game solved as a quantal response equilibrium priced in win probability. On all 164 games of MSI and Worlds 2025 (2,994 team-setups, minimap tracks validated against Riot positions), trailing teams leave the pit empty far more often than leading teams, by +34.8 percentage points at the first baron and +30.4 percentage points at Atakhan. Our model, HASSE-QRE, finds six setup patterns that whole-model series-bootstrap refits reproduce (ARI 0.819, comparable to 0.813 for TAGNet's $K$-Means step alone) and indicates that teams price fights, travel and committed players, not the objective itself (weight 0.018; fixing it to one costs 59.9 log-likelihood points). The equilibrium reproduces a quarter of the commitment gap, almost all of it through state variables rather than its strategic channels. Weighting a team-level contrast by exposure moves it across $p=0.05$ (0.046 against 0.052). We release the tracks, setups and code.

1 Introduction

Professional objective setups are strategically important but have been little measured; this paper models them as player allocations on a walled map and analyzes behavior, structure, and incentives. Across two 2025 events, it finds state-dependent pit commitment, reproducible setup patterns, and an allocation equilibrium in which teams respond more to fight values, travel, and player costs than to the objective’s own value.

1 Introduction

  • Esports analytics had measured little about objective setups, although teams decide which players approach the pit, hold lanes, or wait in the jungle before public spawn times.The closest work, TAGNet, clusters graph-network embeddings of Counter-Strike 2 defenders, but its graph omits map walls, embeds context variables, and compares teams rather than rounds.Source: §1 · p. 2
  • A setup is modeled as an allocation on a walled map, with formations as partitions on Π5, control as a geodesic Voronoi split, and readiness as k-arrival times.The allocation game prices fight and second-front outcomes in win probability.Source: §1 · p. 2
  • +34.8 pp [18.7, 50.0] at the first baron and +30.4 pp [15.0, 46.5] at Atakhan: trailing teams more often leave the pit empty after spawn.The contrasts keep their sign at three pit radii and survive false-discovery control; the pilot dragon gradient disappears when spawn rather than kills anchors the window.Source: §1 · p. 2
  • Six HASSE setup patterns carry the structure, and whole-model series-bootstrap refits agree at ARI 0.819, comparable to TAGNet’s K-Means step at 0.813.Two patterns are spread out at early river objectives, while two gather four players around the pit at late-game spawns.Source: §1 · p. 2
  • LB-QRE estimates the objective-value weight at 0.018 [0.013, 0.026]; fixing it to 1 costs 59.9 log-likelihood points.The equilibrium reproduces about a quarter of the observed commitment gap, almost entirely through state variables rather than strategic channels; the Lanchester and second-front channels are small.Source: §1 · pp. 2–3
  • The analysis covers every game of MSI 2025 and Worlds 2025 under one set of objective rules.The measurement layer is validated against Riot positions, with a median tracking error of 177 map units on a 14,820-unit-wide map.Source: Table 1 · p. 3, §1 · p. 3

2 Data and the Spawn Clock

The study uses two 2025 international events and defines setup snapshots relative to each objective’s rule-based spawn time. This fixed clock avoids anchoring observations to who takes the objective, while parallel video and Riot-position layers trade temporal detail against measurement certainty.

2 Data and the Spawn Clock

  • The core data cover every game of MSI 2025 and Worlds 2025, 164 games in 60 series between 19 teams.An archive of 2,678 professional games is used only to fit outcome models and never sees setup choices.Source: §2 · p. 3
  • Anchoring at the rule-based spawn time avoids selecting a window based on who took the objective; snapshots remain in the risk set only while it is alive.Across 1,516 spawns, no kill precedes the inferred spawn time.Source: §2 · p. 3, Table 2 · p. 4
  • The 2,994 team-setups are sampled every 15 seconds from s−120 to s+180 seconds in a video-track layer and a Riot exact-frame layer.The V layer covers 1,497 spawns with 21 snapshots and a median 90% observed; the R layer has no measurement error but only coarse 60-second frames.Source: §2 · p. 3
  • Gold state is fixed before setup begins, using the gold difference at s−120 seconds on the video layer and s−60 seconds on the Riot layer.LB-QRE uses the last Riot frame at least 30 seconds before spawn; trailing, even, and leading states use cut points ±0.5 after standardization.Source: §2 · p. 4
  • The discovery layer stratifies objectives into early river objectives, Atakhan, and late objectives; descriptive analysis separates respawned dragons and barons.The descriptive groups are separated because their spawn time depends on play.Source: §2 · p. 4

3 The Measurement Layer

The measurement layer converts broadcast video into player positions and theory-defined setup objects, with each step validated against Riot data. Its tracking validation quantifies error and informs robustness checks for later analyses.

3.1 From broadcast to game coordinates

  • A digit classifier reads broadcast clocks at 10 fps, and robust fits across game-time ticks align live segments while excluding replay intervals.All 166 videos align; 23 replay segments are detected and excluded.Source: §3.1 · p. 4
  • Per-game minimap pixels are mapped to game coordinates with an orthographic scale-and-offset transform, registered by correlating the static minimap with the walkable mask.Independent crops agree within 0.04 pixels.Source: §3.1 · p. 4
  • Per-game champion detectors feed a Viterbi tracker that selects detection peaks, held positions, or death states using geodesic-distance movement constraints.Detector scores are calibrated to likelihood ratios, and death states are limited to kill-event death windows.Source: §3.1 · p. 4
  • 177 units median tracker error and 0.785 of errors below 600 units; tracked formations match v5 exactly at 0.538 and by Rand index at 0.849.Validation scores tracks at odd minutes, 60 seconds from the nearest pin; analyses use radius, R-layer, interpolation, and held-position sensitivity checks.Source: §3.1 · p. 5, Figure 2 · p. 5

3.2 Objects defined by theory

  • Formations are connected components of an alive-player proximity graph, represented as partitions on Π5; its lattice metric counts Hasse merges and splits.The proximity threshold is 1,900 units, and dead players are removed rather than imputed.Source: §3.2 · pp. 5–6
  • Geodesic Voronoi cells define team control, while the k-arrival time gives the earliest time a team can have k players at the pit.Pit control is the team’s share of cells within the pit radius.Source: §3.2 · pp. 5–6

4 HASSE-QRE

HASSE-QRE separates setup-pattern discovery from an allocation game explaining player commitments, connecting formations and map control to win-probability-valued choices. HASSE models setup sequences on the partition lattice, while LB-QRE models decisions about pit players, a second front, and timing.

4 HASSE-QRE

  • HASSE-QRE separates pattern discovery from commitment explanation, keeping context variables out of pattern discovery and relating them to its output only afterward.This modular cut means context-pattern associations are not built into the patterns.Source: §4 · p. 6

4.1 HASSE: walks on the partition lattice

  • HASSE represents each setup as observations of alive players, a partition-lattice formation, and each formation block’s zone across 13 snapshots.Patterns have a centre path, rigidity, inertia, and zone preferences; the formation emission uses the lattice metric and restricts formations to alive players.Source: §4.1 · p. 6
  • HASSE models patterns with a sticky Markov chain, where zero switching recovers TAGNet’s one-label-per-setup clustering; EM uses exact forward–backward and Viterbi over 52 centres, plus a concave two-dimensional update.The authors examine K=1–8 using held-out log-likelihood over five series-grouped folds, BIC, and ICL, select no finite K, report K=6, and show K=5 and K=8.Source: §4.1 · p. 7

4.2 LB-QRE: the allocation game

  • LB-QRE values choices using a win-probability surface and models actions as pit commitment, second-front allocation, and whether the pit group arrived by spawn.With five alive players there are 36 actions, and players unable to reach the pit within 60 seconds are infeasible for pit commitment.Source: §4.2 · p. 7
  • The allocation game combines a Lanchester-type fight contest and second-front contest with heterogeneous logit quantal response choices, priced in win-probability points.Choice utilities include objective value, travel time, early positioning, and per-player commitment costs; estimated game parameters are solved by damped fixed-point iteration because the game is not zero-sum when ν differs from one.Source: §4.2 · p. 8
  • The four-channel decomposition evaluates 16 combinations of fight law, payoff curvature, second-front value, and team behaviour, then allocates the equilibrium-gap change with Shapley values.The decomposition expresses each channel in the same unit: the difference in equilibrium probability of committing at least three players between trailing and leading teams.Source: §4.2 · pp. 8–9

5 Results

Across the 2025 MSI and Worlds games, trailing teams leave objective pits empty more often after spawns. The results also describe reproducible setup patterns and find that commitment responds mainly to state variables, fight values, travel, and player costs.

5.1 Trailing teams leave the pit empty

  • +34.8 pp at the first baron and +30.4 pp at Atakhan: trailing teams more often leave the pit empty after the spawn than leading teams.The contrast appears across objectives and time; headline cells have the same sign at all three pit radii and pass false-discovery control.Source: §5.1 · p. 9, Figure 6 · p. 10

5.2 The estimator decides what is significant

  • −6.4 pp (p=0.046) for unweighted team means becomes −4.1 pp (p=0.052) when weighted by exposure, placing the significance threshold near the estimator choice.The estimates are within about ±0.007 bootstrap Monte Carlo noise; the spawn-anchored first-dragon contrast does not reproduce the kill-anchored pilot gradient.Source: §5.2 · pp. 10–11, Figure 7 · p. 10

5.3 Arriving first decides most river objectives

  • 30 s after spawn is the median first arrival; the first team to bring three players reaches the objective within 180 s in 0.68–0.80 of river-objective cases, but 0.51 of Atakhans.The stated proportions are 0.80 for respawned dragons, 0.77 for heralds, 0.68 for first barons, and 0.51 for Atakhans.Source: §5.3 · p. 11

5.4 Six patterns, reproducible

  • K=6 is a convenient level of detail, not a criterion-selected count; its six patterns distinguish early spread formations from late setups that gather four players at the pit.Patterns 1 and 4 are rigid and concentrated in early river objectives; patterns 3 and 5 dominate late objectives and are less rigid and inert. The pattern guide encodes paths on Π5 and objective-class mixture weights.Source: §5.4 · pp. 11–12, Figure 8 · p. 11

5.5 What context does to patterns

  • A one-standard-deviation gold lead raises the gathered late pattern 5 by +0.024, while holding the dragon-soul point raises pattern 4 by +0.030.Twenty-four of 48 context effects have intervals excluding zero; Figure 10 compares next-death probabilities by local numbers with a stochastic square law and shows Tullock ratios under two count definitions.Source: §5.5 · pp. 12–13, Figure 10 · p. 13

5.6 Fight laws

  • At 5v3, the larger side loses the next player with probability 0.037, versus 0.375 under the square law.The fitted Tullock fight law on tracked counts has γ_n=0.79 and γ_g=10.9; adding participants from v5 is treated as a sensitivity.Source: §5.6 · p. 13

5.7 What the equilibrium reproduces

  • The observed commitment gap is −19.2 pp, while LB-QRE produces −4.7 pp, about 0.24 of the observed gap; with all channels off, it is still −5.1 pp.The four channels add +0.5 pp net: fight-probability curvature and team behaviour pull in opposite directions, while Lanchester and second-front channels are small. Figure 11 displays the gap, channel decomposition, and loss from fixing ν=1; only 57% of bootstrap replicates meet the stated convergence tolerance.Source: §5.7 · p. 14, Figure 11 · p. 14

5.8 Team fingerprints are mostly noise

  • Team usage is noisy: split-half correlations range from −0.04 to 0.35, and intervals include zero for five of six patterns.At 31 to 394 setups per team, patterns describe the league but do not yet separate teams; Figure 12 shows usage intervals, split-half reliability, and decision-precision intervals.Source: §5.8 · p. 14, Figure 12 · p. 15

6 Related Work

The paper positions its approach against work on esports tactics, spatial formations, and equilibrium in sport. It replaces learned representations and geometric templates with operational map-based allocations and combines established game-theoretic components.

Tactics in esports

  • TAGNet is the closest esports-tactics study; this work replaces its learned embedding with objects that have units, its fixed K with held-out evidence, and its context-in-representation design with a cut.The paper also situates itself alongside work on Dota 2 team spread, Counter-Strike action values and trajectories, and LoL prediction, rating, networks, and performance indicators.Source: §6 · p. 15

Space and formations in team sports

  • Unlike geometric-template approaches to formations, the paper treats formations as set partitions and counts merges and splits with a lattice metric.It describes HASSE as a moving centre-partition path with rigidity and inertia inside a sticky HMM, and lets evidence rather than a fixed K determine pattern count.Source: §6 · p. 15

Game theory in sport

  • The explanation layer combines quantal response equilibrium, Lanchester–Tullock fights, and a Blotto second front, with Shapley decomposition comparing their contributions.The cited foundations include equilibrium tests in sport and models of risk-taking by trailing contestants.Source: §6 · pp. 15–16

7 Limitations

The evidence is constrained by imperfect minimap measurements, unobserved game state, and coverage of one ruleset across two international events.

Limitations

  • Behavioural claims are conditional on observed state: vision, cooldowns, wave states, and voice communication are unobserved, and the data cover one ruleset and two events.Win probability and event values are outcome-only associations from the archive, not causal effects.Source: §7 · p. 16

8 Conclusion

The conclusion frames objective setups as player allocations on a walled map and reports recurring post-spawn behaviour, reproducible partition-lattice patterns, and commitment tied more to fights, travel, and player costs than to the prize. It says these objects can also apply to other settings where teams divide players between a contested point and the rest of the map.

Conclusion

  • Measured as allocations on a walled map, setups show post-spawn differences by gold state, reproducible partition-lattice walks, and commitment that responds more to fights, travel, and player costs than to the prize.The conclusion states that these findings do not require a neural embedding.Source: §8 · p. 16
  • The same objects apply to settings where teams split players between a contested point and the rest of the map, including other MOBAs, tactical shooters, and team sports with set pieces.The tracks, setups, and code are released with the paper.Source: §8 · p. 16

A Proofs

The appendix proves the partition-lattice distance used to count formation changes and establishes the equilibrium result for the two-team choice model. It also specifies HASSE estimation and reports recovery checks on synthetic data.

A Proofs

  • The Π5 distance equals shortest-path length on the Hasse diagram, with each shortest path decomposing into a fixed number of merges and splits.The proof reports exhaustive verification over all partition pairs and triples.Source: A Proofs · p. 19
  • The contesting-versus-conceding payoff difference is S(p − π*); under logit choice, the contesting probability is Λ(λ times that difference).Source: A Proofs · p. 19
  • With ν = 1, the two teams’ win-probability changes are opposites, and their logit best responses correspond to a unique saddle point.Source: A Proofs · p. 19

B HASSE: Estimation

  • HASSE estimation uses exact forward–backward inference, then updates pattern centres by Viterbi and optimizes the remaining pattern and zone parameters.The centre update penalizes one Hasse step between consecutive centres; the other updates include a concave likelihood and penalized multinomial logit.Source: B HASSE: Estimation · p. 19
  • Synthetic-data recovery found identical log-likelihoods across restarts, exact recovery of all three planted centre paths, and first-snapshot assignment ARI 0.915.With 40% of setups missing the last snapshot, assignment ARI was 0.903.Source: B HASSE: Estimation · p. 19

C LB-QRE: Specification

The specification defines when commitment is measured, restricts which players can reach the pit, and prices objective, fight, and tower outcomes using win-probability values. It estimates choice and strategy parameters iteratively, with tracking error reported against Riot positions.

C LB-QRE: Specification

  • Commitment time is the first at-risk snapshot from spawn to 180 seconds when either team has at least three players near the pit, or the last at-risk snapshot otherwise.Players more than 60 seconds of movement away at spawn are infeasible for the pit.Source: C LB-QRE: Specification · pp. 19–20
  • The event-value table reports values in win-probability points at W = 0.5, with a separate total-effect measure based on the result.Source: Table 4 · p. 20
  • Tracker quality is assessed against Riot V5 positions at held-out odd minutes across 23,286 champion-minutes and 164 games.The caption specifies map units and distinguishes crowded from uncrowded cases.Source: Table 5 · p. 20
  • Objective, fight, and tower values are read from event regressions evaluated at the out-of-fold win probability before spawn.Source: C LB-QRE: Specification · p. 20
  • Estimation alternates conditional-logit parameter maximization with strategy updates, shrinking team deviations toward global values and pooling teams with too few contests.Source: C LB-QRE: Specification · p. 20

E T1 and Faker

T1’s pattern usage and commitment behavior resemble the league, while Faker arrives closer to the pit and earlier than other starting mid laners. The analyses also identify limits on what position-based measures can establish and exclude earlier Worlds rule sets.

E T1 and Faker

  • T1’s six-pattern usage places it near Gen.G, but low split-half reliability means the fingerprints do not establish a distinct team identity.Gen.G is T1’s nearest team by total-variation distance, at 0.032.Source: E T1 and Faker · p. 21
  • Faker is 469 units closer to the pit before spawn and arrives earlier in his team’s order, but he is no more often central or in the main formation block.The comparison is against other starting mid laners.Source: E T1 and Faker · p. 21
  • T1’s trailing-minus-leading commitment contrast is −0.194, versus −0.192 for the rest of the league; their difference is −0.002.The reported intervals for the difference include zero.Source: E T1 and Faker · p. 21
  • Across 399 T1 fights, the mean fight-law residual is −0.002, consistent with the law’s prediction at equal numbers and gold.Source: E T1 and Faker · p. 22
  • T1 committed at least three players in 0.203 of contests at MSI and 0.254 at Worlds 2025.Source: E T1 and Faker · p. 23
  • Position-based measures do not cover champion-pool depth, draft play, or communication, and Worlds 2023–2024 setups are outside the study because objective rules differed.Source: E T1 and Faker · p. 23

F Dataset

The release packages data products for the 164 core games, including aligned clocks, registered minimaps, labeled player tracks, setup records, and the navigation graph. It also includes code that produces every figure and table.

F Dataset

  • The release includes clock alignments per video and minimap registration parameters for the 164 core games.Source: F Dataset · p. 23
  • Released 2-fps tracks label each position by source, including detected, interpolated, held, pinned, and dead states.Source: F Dataset · p. 23
  • The release provides 1,516 spawn records and 2,994 team-setups on a 15-second grid, with partitions, zones, arrival measures, pit control, and the navigation graph.Source: F Dataset · p. 23
  • Code producing every figure and table is released with the data products.Source: F Dataset · p. 23
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